Thursday, December 17, 2009

Q: Name five two-digit numbers that are evenly spaced out — like 32, 34, 36, 38 and 40 — in which all 10 digits from 0 to 9 are used once each. What numbers are these?

The first answer came to me as I was driving and listening to the puzzle. However, as I was typing this clue, I came up with another answer that also works. How to give a clue without giving it away, I don't know.

Edit: The clues were to driving which should have made you think of either a car (ie. Car 54) or golf (ie. 18 holes). Those were the starting numbers of two of the four possible sequences.

The tens digit will either step by 1 (5,6,7,8,9) or by 2 (1,3,5,7,9) always ending on 9. The ones digit will also step by 1 (0,1,2,3,4) or by 2 (0,2,4,6,8) always starting (or ending) on 0. That leads to 4 possible sequences:
Starting with 10 stepping by 22 (20+2)
Starting with 18 stepping by 18 (20-2)
Starting with 50 stepping by 11 (10+1)
Starting with 54 stepping by 9 (10-1)

61 comments:

Here's my standard reminder... don't post the answer or any outright spoilers before the deadline of Thursday at 3pm ET. If you know the answer, click the link and submit it to NPR, but don't give it away here. Thank you.

Blaine, I believe there are exactly three possible correct answers. However, my method could not have been used while multi-tasking in the car. (I think that's called "distracted driving".)In hindsight, I see how two of the answrs might have come to mind relatively quickly (but not to my mind). The third answer has a more complicated pattern.

Thanks, Les, for posting this week’s puzzle a day early. Last night, I uncorked a bottle of wine, put on my favorite recording of Eine Kleine Nachtmusik, and had the (an) answer in a few minutes. No computer programs here – just a ball-point pen and a piece of paper.

Chuck,While you were listening to Eine Kleine Nachtmusik, I was stressing out driving and listening to my teenagers argue about what radio station we'd listen to when it suddenly came to me in a Halloweenish vision.

In standard western math notation, leading zeros are insignificant and therefore truncated. I.e., “01” is actually "1" and is therefore not a bona-fide two-digit number. “000001” also equals "1" and is not a 6-, 5-, 4-, 3- or 2-digit number. So my two-cents is that anybody who has a starting number of less than 10 is not going to get on Will’s scoreboard this week :)

If the answers of the most obvious ,IMHO, answer are written in order in a column, a circular path becomes apparent. If the column is totaled, another number relating to a circle is produced which can lead to the middle term. For some puzzles I just don't care how much I give away. They're just too easy to begin with.

I suspect the alternate answers are more interesting than the intended one. I'll work on them when I get home.

There are 4 answers if you don't count leading 0s, 8 if you do. Two (four with leading zeros) were obvious to me. One of the others is obvious in retrospect, but wasn't immediately. The final one is less obvious.

I wrote a little computer program in Qbasic and confirmed that Blaine and David Stewart were right. Using only numbers that would commonly be regarded as 2-digits (10 through 99), there are four sequences that fit the bill.

phredp, interesting observation. In fact, of the four possible solutions, two have your middle number in common while the other two have a different middle number in common. So you now know the middle number in each of the four solutions.

Now I (finally ) found the four solutions, the symmetry is very nice (increments are a, b, 2a and 2b)Anyone else think the puzzle was redundantly worded: "Name 5 numbers" and "What numbers are these" ? Jeez we're a picky bunch...

I stopped at two answers. I frankly wish I had waited to send in my answer since the second one I came up with is somewhat more elegant than the first one I came to. But this is a challenge that doesn't depend on inspiration to strike you, it can be done relatively easily with a process of elimination. And if you're bad with math and can't get it that way, WS seemed to give you instructions on how to go out and buy the answer this week. I suspect we'll have a record number of entries, even if Sunday Puzzle fans are more verbal than mathematically inclined.

1. In the dictionary.2. The letter D.3. On the outside.4. Sandy claws ("Santa Claus").5. A Snow Ball.6. Neither, they both burn shorter.7. Lunch and dinner.8. One, because then it's not empty.9. ?10. When it is frosted.11. An umbrella.12. To keep his hands warm.13. Ohio ("O-high-O").14. ?15. ?

Ken, I just went back to Chuck's puzzle and you're absolutely right, but it was totally unintentional. That answer must have been floating around somewhere in my brain when I thought this puzzle up. Nice job.

Heres another one that I just thought up:

The nickname of a famous queen is an anagram of the name of a famous king. What are their names?

Congrats to all who participated in the Stupid Christmas Quiz! They all got solved.

Here’s another quiz that will get your puzzle juices going...

The How Many Quiz

Example: 60 M in an H = 60 Minutes in an Hour

1. 1 W on a U2. 3 B M (S H T R)3. 3 W on a T4. 4 Q in a G5. 5 (or 9) D in a Z C6. 7 W of the W7. 8 A on an O8. 8 (or 9) P in the S S9. 8 S on a S S10. 9 L of a C11. 11 P on a F T12. 12 S of the Z13. 13 in a B D14. 13 S on the A F15. 18 H on a G C16. 24 H in a D17. 26 L in the A18. 29 D in F in a L Y19. 32 D F at which W F20. 40 D and N of the G F21. 54 C in a D (with the J)22. 57 H V23. 64 S on a C24. 66 B in the B25. 88 P K26. 90 D in a R A27. 1,000 (or 10,000) W that a P is W28. 1,001 A N

I tried to disguise the answers by printing them upside down, but perhaps your system doesn't have the right Unicode font. Don't read the following if you are still working on the puzzle.

1. 1 Wheels on a Unicycle2. 3 Blind Mice (See How They Run)3. 3 Wheels on a Tricycle4. 4 Quarts in a Gallon5. 5 (or 9) Digits in a Zip Code6. 7 Wonders of the World7. 8 Arms on an Octopus8. 8 (or 9) Planets in the Solar System9. 8 Sides on a Stop Sign10. 9 Lives of a Cat11. 11 Players on a Football Team12. 12 Signs of the Zodiac13. 13 in a Baker's Dozen14. 13 Stripes on the American Flag15. 18 Holes on a Golf Course16. 24 Hours in a Day17. 26 Letters in the Alphabet18. 29 Days in February in a Leap Year19. 32 Degrees Farenheit at which Water Freezes20. 40 Days and Nights of the Great Flood21. 54 Cards in a Deck (with the Jokers)22. 57 Heinz Varieties23. 64 Squares on a Chessboard (or Checkerboard)24. 66 Books in the Bible25. 88 Piano Keys26. 90 Degrees in a Right Angle27. 1,000 (or 10,000) Words that a Picture is Worth28. 1,001 Arabian Nights

The following is a quiz put together by some MENSA women. It was published a number of years ago in PLAYBOY Magazine, of all places. Seems they did a nude / semi-nude shoot of dozen or so MENSA lovelies a while back and the quiz added a "beauty and brains" dimension to the piece. Anyway, I'm upping it here for your enjoyment. I found a few of the questions to be relatively easy. But some were tough and some very tough. Good luck :)

1. I am a man. If Joe's son is my son's father, what relationship am I to Joe?

a) His grandfatherb) His fatherc) His sond) His grandsone) I am Joef) His uncle

2. Find the number that logically completes the series:

2, 3, 5, 9, 17, ...

3. What number comes next in this series:

9, 16, 25, 36, ...

4. Complete this analogy with a 5-letter word ending with the letter H.

High is to low as sky is to _ _ _ _ H.

5. In the grid below, two of the numbers in a line (across and down) produce the third. What is the missing number?

6 2 42 ? 04 0 4

6. Complete this analogy with a 7-letter word ending with the letter T.

Potential is to actual as future is to _ _ _ _ _ _ T.

7. If it were two hours later, it would be half as long until midnight as it would be if it were an hour later. What time is it now?

8. Statistics indicate that male drivers are involved in more accidents than female drivers. The only conclusion that can properly be drawn is that:

a) Male chauvinists are wrong, as usual, about women's abilities.b) Men are actually better drivers but drive more frequently.c) Men and women drive equally well, but men log more total mileage.d) Most truck drivers are male.e) There is not enough information to justify a conclusion.

9. In the following set of numbers, a rule of arithmetic applies across and down so that two of the numbers in a line produce the third. What is the missing number?

11. What word do the following letters make when they are unscrambled?

L E L E I N S O V S

12. Find the two words nearest in meaning to each other.

a) beamb) lumpc) giggled) raye) collection

13. If Jim turns right or left at the stop sign, he will run out of gas before he reaches a service station. He has already gone too far past a service station to return before he runs out of gas. He does not see a service station ahead. Which one of the following statements can be positively deduced:

a) He may run out of gas.b) He will run out of gas.c) He should not have taken this route.d) He is lost.e) He should turn right at the stop sign.f) He should turn left at the stop sign.

14. If an airplane travels at an average rate of 500 MPH, how long will it take to complete 20 trips, of which 5 are for 1000 miles, 5 are for 1500 miles, 5 are for 2000 miles and 5 are for 3000 miles?

1. There's one sport in which neither the spectators nor the participants know the score or the leader until the contest ends. What is it?

2. What famous North American landmark is constantly moving backward?

3. Of all vegetables, only a few can live to produce on their own for several growing seasons. All other vegetables must be replanted every year. What are the perennial vegetables?

4. At noon and midnight the hour and minute hands are exactly coincident with each other. How many other times between noon and midnight do the hour and minute hands cross?

5. What is the only sport in which the ball is always in the possession of the team on defense, and the offensive team can score without touching the ball?

6. What fruit has its seeds on the outside?

7. In many liquor stores, you can buy pear brandy, with a real pear inside the bottle. The pear is whole and ripe, and the bottle is genuine; it hasn't been cut in any way. How did the pear get inside the bottle?

8. Only three words in standard English begin with the letters "dw." They are all common. Name them. (No dictionaries now!)

9. There are fourteen or fifteen punctuation marks in English grammar. Can you name them?

10. Where are the lakes that are referred to in the "Los Angeles Lakers?"

11. There are seven ways a baseball player can legally reach first base without getting a hit. Taking a base on balls, a walk, is one way. Name the other six.

12. It's the only vegetable or fruit that is never sold frozen, canned, processed, cooked, or in any other form but fresh. What is it?

13. Name ten or more things that you can wear on your feet that begin with the letter "s."

[(1)(2)] The 9's multiplication table occurred too easily to meand contained (2), my solution to the puzzle. I did fiddle around later to see how the other solutions found by other morepersistent solvers might be related.

(3)(4) Inverting the units columns of (1) and (2)provides 2 more solutions.

(5)(6) Since [(1)(2)] together are a series, two more solutions can be formed by using every other value.(The same simple thing can't be done with non series [(3)(4)].)

(7)(8) Inverting the units columns of (5) and (6)provides 2 more solutions.

If memory serves, I don’t think the MENSA quiz was timed. It was just to see if you could answer them at all. As for the most recent quiz:

1 – OK2 – Still open for someone.3 – Still open for someone.4 – OK5 – Not the answer I was looking for. Still open for someone.6 – OK7 – OK8 – OK9 – You got many but there are still at least four open for someone. BTW, I don’t believe the ampersand, @ or underscore are considered punctuation marks.10 – OK11 – You got most but there’s one still open for someone. BTW, the catcher dropping a third strike doesn’t automatically get you on. You still have to “steal” first, i.e., get there.12 – Not the answer I was looking for. Still open for someone.13 – You got many but there are still at least four open for someone. BTW, I don’t think I’d count shoe brands, a particular type of skin or sweat socks (in addition to socks). But that’s just me.

Good job!

I found a couple more oldies but goodies, but I think I’ll wait ‘til next week to post them. Meanwhile, let’s get this one put to bed.

11. Interference by a defensive player. In the interest of brevity, I didn't want to elaborate on the dropped third strike by the catcher since it's a bit complicated, but I'm aware that the batter still has to reach first before he/she is either tagged out or a player touches first while he/she is possessing the ball.

My reasoning was that it makes sense to think of the number as two digits, not a single number. Then you need to increment or decrement each digit during each iteration. Adding 9 to a two digit number will increase the 10's digit by one while decreasing the 1's digit by one. Adding 11 will increase both digits by 1. Those are the two obvious ones. Less obvious initially, to me, was adding 22 will increase both digits by 2 and adding 18 will increase the 10's digit by 2 while decreasing the 1's digit by 2. Using this you can easily construct the proper sequences of numbers by picking appropriate starting numbers.

My first thought went to multiples of nines (54, 63, 72, 81, 90) and eighteen (18, 36, 54, 72, 90) where the tens digit went up to 9 and the ones digit went down to 0. I shortly realized you could create a similar sequence where both digits went up by either 1 (50, 61, 72, 83, 94) or 2 (10, 32, 54, 76, 98).

So I wonder which of these was "the" set of numbers that is printed in Scott Kim's calendar? Anyone else disappointed that the puzzle given as a reason to buy the calendar doesn't have a unique answer? :-)

2. Right – Niagara Falls.3. Right – asparagus is one. There are still two more perennial vegetables.5. Right – baseball.9. Right – dash, slash and brackets are all good. There is still one more punctuation mark.11. There is still one more way to get on first.12. The only-served-fresh vegetable or fruit I have in mind remains unnamed.

As to Will’s puzzle:

I took a different tack to solve it: a tiny little computer program written in QBasic. For those of you using Windows as your operating system you probably already have QBasic on your C: drive and never noticed it. It’s almost always on the original installation discs. Anyway, if anyone has QBasic and would like to see and run the program, it’s only about 50 lines of code. I’ll be happy to post it. It solves the puzzle essentially instantaneously.

I am stunned with the immediacy of the Internet and a somewhat amazing coincidence that just occurred!

You remember the How Many Quiz I recently posted here that begins 1 W on a U, 3 B M (S H T R), etc? I had originally run across it decades ago. It’s so old it looks typewritten, not word-processed. I long ago lost my copy but fortunately, I had given a copy to one of the secretaries at work. She still works for me and she still had her copy. We got it out, went over it and added a few new items. Then I typed it into Word, edited some of the clues and changed the order the entries were in. Finally, I posted it on the puzzle blog. The date and time stamp of my posting was December 16, 6:54PM PST.

Last night, I got an e-mail from my daughter. She was forwarding me an e-mail she had received from an acquaintance. The date and time stamp of her friend’s e-mail is December 16, 9:04 PM PST. It contained the exact same quiz – word for word, letter for letter – that I had posted a little over 2 hours earlier!!!

What are the chances of that happening? And I wonder how many degrees of separation there were.

P.S. From deep in my C: drive’s sub-directories and the distant past, I’ve restored a little different kind of word quiz for next week :)

P.P.S. My daughter responded to the guy who sent her the How Many Quiz. By amazing coincidence, he liked it, thought she’d be interested and sent it to her. He had no idea that “Chuck” was the father of the person he’s dated. Small world!

"Think of a familiar two-word phrase, five letters in each word. The second word starts with P. The phrase names something that is nice to have after dinner. Change the P to an S, and you'll get another familiar phrase that names something that's nice to have before you start a job. What phrases are these?"